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Probability Tales
Page60(71 of 250)

60 1. Streaks 0 0.2 0.4 0.6 0.8 Point probability 0.2 0.4 0.6 0.8 1 Set probability Figure 27. Set-winning probabilities for players of equal abilities reason that we need two winning states for each player is that we need to keep track of who serves first in the subsequent set (if there is one). The transition probabilities are found by using the graph shown in Figure 25; for example, if the first player is serving, the game score is 2-1, and he has a probability of .6 of winning a given point, then the game score will become 3-1 with probability .74. Once again, we are interested in the probability that the player who serves first wins the set. This time, there are two parameters, namely the probabilities that each of the players wins a given point when they are serving. We denote these two parameters by p1 and p2. If we let p1 = p2, then Figure 27 shows the probability that the first player wins the set as a function of p1. There is nothing very remarkable about this graph. Suppose instead that p1 = p2 + .1, i.e. the first player is somewhat better than the second player. In this case, Figure 28 shows the probability that the first player wins the set as a function of p1 (given that the first player serves the first game). If, for example, p1 = .55 and p2 = .45, then the probability that the first player wins the set is .81.